The 21-year-old struggled to break into the first team at Goodison Park after spending five years in the Merseyside outfit's academy.
Last season, the goalkeeper spent four months on loan at Motherwell, making four appearances, and after being released by Everton he has received a permanent offer from the Scottish club.
"Russell showed a tremendous attitude towards the tail end of last season and put in some terrific performances for us during an important run of games, manager Stephen Robinson told the club's official website.
"We're delighted to have secured Russell's signature and to have that competition for the number one jersey is crucial."
Griffiths failed to make a single first-team appearance for Everton.